Divisibility by 3 of even multiperfect numbers of abundancy 3 and 4
Broughan, K. A. & Zhou, Q. (2010). Divisibility by 3 of even multiperfect numbers of abundancy 3 and 4. Journal of Integer Sequence, 13, 10.1.5
Permanent Research Commons link: https://hdl.handle.net/10289/3623
We say a number is flat if it can be written as a non-trivial power of 2 times an odd squarefree number. The power is the “exponent” and the number of odd primes the “length”. Let N be flat and 4-perfect with exponent a and length m. If a ≢ 1 mod 12, then a is even. If a is even and 3 ∤ N then m is also even. If a ≡ 1 mod 12 then 3 | N and m is even. If N is flat and 3-perfect and 3 ∤ N, then if a a ≡ 1 mod 12, a is even. If a ≡ 1 mod 12 then m is odd. If N is flat and 3 or 4-perfect then it is divisible by at least one Mersenne prime, but not all odd prime divisors are Mersenne. We also give some conditions for the divisibility by 3 of an arbitrary even 4-perfect number.
This article has been published in the journal: Journal of Integer Sequence. ©2010 K. A. Broughan & Q. Zhou.